let Kb, Cb be Subset of (TOP-REAL 2); :: thesis: ( Kb = { p where p is Point of (TOP-REAL 2) : ( not - 1 <= p `1 or not p `1 <= 1 or not - 1 <= p `2 or not p `2 <= 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| > 1 } implies Sq_Circ .: Kb = Cb )
assume A1: ( Kb = { p where p is Point of (TOP-REAL 2) : ( not - 1 <= p `1 or not p `1 <= 1 or not - 1 <= p `2 or not p `2 <= 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| > 1 } ) ; :: thesis: Sq_Circ .: Kb = Cb
thus Sq_Circ .: Kb c= Cb :: according to XBOOLE_0:def 10 :: thesis: Cb c= Sq_Circ .: Kb
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in Sq_Circ .: Kb or y in Cb )
assume y in Sq_Circ .: Kb ; :: thesis: y in Cb
then consider x being set such that
x in dom Sq_Circ and
A2: x in Kb and
A3: y = Sq_Circ . x by FUNCT_1:def 12;
consider q being Point of (TOP-REAL 2) such that
A4: q = x and
A5: ( not - 1 <= q `1 or not q `1 <= 1 or not - 1 <= q `2 or not q `2 <= 1 ) by A1, A2;
now
per cases ( q = 0. (TOP-REAL 2) or ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1 ) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1 ) ) ) ) or ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1 ) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1 ) ) ) ) ;
case A6: ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1 ) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1 ) ) ) ) ; :: thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| > 1 )
then A7: Sq_Circ . q = |[((q `1 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| by JGRAPH_3:def 1;
A8: ( ( - 1 <= q `2 & q `2 <= 1 ) or - 1 > q `1 or q `1 > 1 )
proof
assume A9: ( not - 1 <= q `2 or not q `2 <= 1 ) ; :: thesis: ( - 1 > q `1 or q `1 > 1 )
now
per cases ( - 1 > q `2 or q `2 > 1 ) by A9;
case A10: - 1 > q `2 ; :: thesis: ( - 1 > q `1 or q `1 > 1 )
then ( - (q `1 ) < - 1 or ( q `2 >= q `1 & q `2 <= - (q `1 ) ) ) by A6, XXREAL_0:2;
hence ( - 1 > q `1 or q `1 > 1 ) by A10, XREAL_1:26, XXREAL_0:2; :: thesis: verum
end;
case q `2 > 1 ; :: thesis: ( - 1 > q `1 or q `1 > 1 )
then ( 1 < q `1 or 1 < - (q `1 ) ) by A6, XXREAL_0:2;
then ( 1 < q `1 or - (- (q `1 )) < - 1 ) by XREAL_1:26;
hence ( - 1 > q `1 or q `1 > 1 ) ; :: thesis: verum
end;
end;
end;
hence ( - 1 > q `1 or q `1 > 1 ) ; :: thesis: verum
end;
A11: |[((q `1 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `1 = (q `1 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) by EUCLID:56;
A12: |[((q `1 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]| `2 = (q `2 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) by EUCLID:56;
A13: 1 + (((q `2 ) / (q `1 )) ^2 ) > 0 by XREAL_1:36, XREAL_1:65;
then A16: (q `1 ) ^2 > 0 by SQUARE_1:74;
(q `1 ) ^2 > 1 ^2 by A5, A8, SQUARE_1:117;
then A17: sqrt ((q `1 ) ^2 ) > 1 by SQUARE_1:83, SQUARE_1:95;
|.|[((q `1 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| ^2 = (((q `1 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) + (((q `2 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) by A11, A12, JGRAPH_3:10
.= (((q `1 ) ^2 ) / ((sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))) ^2 ) by XCMPLX_1:77
.= (((q `1 ) ^2 ) / ((sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / ((sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by XCMPLX_1:77
.= (((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / ((sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))) ^2 )) by A13, SQUARE_1:def 4
.= (((q `1 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `2 ) / (q `1 )) ^2 ))) by A13, SQUARE_1:def 4
.= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) / (q `1 )) ^2 )) by XCMPLX_1:63
.= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `2 ) ^2 ) / ((q `1 ) ^2 ))) by XCMPLX_1:77
.= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `1 ) ^2 )) + (((q `2 ) ^2 ) / ((q `1 ) ^2 ))) by A16, XCMPLX_1:60
.= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `1 ) ^2 )) by XCMPLX_1:63
.= ((q `1 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 ))) by XCMPLX_1:82
.= ((q `1 ) ^2 ) * 1 by A14, COMPLEX1:2, XCMPLX_1:60
.= (q `1 ) ^2 ;
then |.|[((q `1 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `2 ) / (q `1 )) ^2 ))))]|.| > 1 by A17, SQUARE_1:89;
hence ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| > 1 ) by A3, A4, A7; :: thesis: verum
end;
case A18: ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1 ) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1 ) ) ) ; :: thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| > 1 )
then A19: Sq_Circ . q = |[((q `1 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| by JGRAPH_3:def 1;
A20: |[((q `1 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `1 = (q `1 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) by EUCLID:56;
A21: |[((q `1 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]| `2 = (q `2 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) by EUCLID:56;
A22: 1 + (((q `1 ) / (q `2 )) ^2 ) > 0 by XREAL_1:36, XREAL_1:65;
A23: q `2 <> 0 by A18;
then A24: (q `2 ) ^2 > 0 by SQUARE_1:74;
( ( - 1 <= q `1 & q `1 <= 1 ) or - 1 > q `2 or q `2 > 1 )
proof
assume A25: ( not - 1 <= q `1 or not q `1 <= 1 ) ; :: thesis: ( - 1 > q `2 or q `2 > 1 )
now
per cases ( - 1 > q `1 or q `1 > 1 ) by A25;
case A26: - 1 > q `1 ; :: thesis: ( - 1 > q `2 or q `2 > 1 )
then ( q `2 < - 1 or ( q `1 < q `2 & - (q `2 ) < - (- (q `1 )) ) ) by A18, XREAL_1:26, XXREAL_0:2;
then ( - (q `2 ) < - 1 or - 1 > q `2 ) by A26, XXREAL_0:2;
hence ( - 1 > q `2 or q `2 > 1 ) by XREAL_1:26; :: thesis: verum
end;
case A27: q `1 > 1 ; :: thesis: ( - 1 > q `2 or q `2 > 1 )
( ( - (- (q `1 )) < - (q `2 ) & q `2 < q `1 ) or ( q `2 > q `1 & q `2 > - (q `1 ) ) ) by A18, XREAL_1:26;
then ( 1 < - (q `2 ) or ( q `2 > q `1 & q `2 > - (q `1 ) ) ) by A27, XXREAL_0:2;
then ( - 1 > - (- (q `2 )) or 1 < q `2 ) by A27, XREAL_1:26, XXREAL_0:2;
hence ( - 1 > q `2 or q `2 > 1 ) ; :: thesis: verum
end;
end;
end;
hence ( - 1 > q `2 or q `2 > 1 ) ; :: thesis: verum
end;
then (q `2 ) ^2 > 1 ^2 by A5, SQUARE_1:117;
then A28: sqrt ((q `2 ) ^2 ) > 1 by SQUARE_1:83, SQUARE_1:95;
|.|[((q `1 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| ^2 = (((q `1 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) + (((q `2 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) by A20, A21, JGRAPH_3:10
.= (((q `1 ) ^2 ) / ((sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))) ^2 ) by XCMPLX_1:77
.= (((q `1 ) ^2 ) / ((sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) + (((q `2 ) ^2 ) / ((sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by XCMPLX_1:77
.= (((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / ((sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))) ^2 )) by A22, SQUARE_1:def 4
.= (((q `1 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) + (((q `2 ) ^2 ) / (1 + (((q `1 ) / (q `2 )) ^2 ))) by A22, SQUARE_1:def 4
.= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) / (q `2 )) ^2 )) by XCMPLX_1:63
.= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (1 + (((q `1 ) ^2 ) / ((q `2 ) ^2 ))) by XCMPLX_1:77
.= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) / ((q `2 ) ^2 )) + (((q `2 ) ^2 ) / ((q `2 ) ^2 ))) by A24, XCMPLX_1:60
.= (((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / ((q `2 ) ^2 )) by XCMPLX_1:63
.= ((q `2 ) ^2 ) * ((((q `1 ) ^2 ) + ((q `2 ) ^2 )) / (((q `1 ) ^2 ) + ((q `2 ) ^2 ))) by XCMPLX_1:82
.= ((q `2 ) ^2 ) * 1 by A23, COMPLEX1:2, XCMPLX_1:60
.= (q `2 ) ^2 ;
then |.|[((q `1 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 )))),((q `2 ) / (sqrt (1 + (((q `1 ) / (q `2 )) ^2 ))))]|.| > 1 by A28, SQUARE_1:89;
hence ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| > 1 ) by A3, A4, A19; :: thesis: verum
end;
end;
end;
hence y in Cb by A1; :: thesis: verum
end;
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in Cb or y in Sq_Circ .: Kb )
assume y in Cb ; :: thesis: y in Sq_Circ .: Kb
then consider p2 being Point of (TOP-REAL 2) such that
A29: p2 = y and
A30: |.p2.| > 1 by A1;
set q = p2;
now
per cases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1 ) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1 ) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1 ) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1 ) ) ) ) ;
case A31: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1 ) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1 ) ) ) ) ; :: thesis: ex x being set st
( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x )
set px = |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]|;
A32: |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 = (p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) by EUCLID:56;
A33: |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 = (p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) by EUCLID:56;
1 + (((p2 `2 ) / (p2 `1 )) ^2 ) > 0 by XREAL_1:36, XREAL_1:65;
then A34: sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )) > 0 by SQUARE_1:93;
A35: 1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ) > 0 by XREAL_1:36, XREAL_1:65;
A36: (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) = (p2 `2 ) / (p2 `1 ) by A32, A33, A34, XCMPLX_1:92;
A37: p2 `1 = ((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) / (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) by A34, XCMPLX_1:90
.= (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) / (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) by EUCLID:56 ;
A38: p2 `2 = ((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))) / (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) by A34, XCMPLX_1:90
.= (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) by EUCLID:56 ;
A39: |.p2.| ^2 = ((p2 `1 ) ^2 ) + ((p2 `2 ) ^2 ) by JGRAPH_3:10;
A40: |.p2.| ^2 > 1 ^2 by A30, SQUARE_1:78;
A41: now
assume that
A42: |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 = 0 and
A43: |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 = 0 ; :: thesis: contradiction
A44: (p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) = 0 by A42, EUCLID:56;
A45: (p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) = 0 by A43, EUCLID:56;
A46: p2 `1 = 0 by A34, A44, XCMPLX_1:6;
p2 `2 = 0 by A34, A45, XCMPLX_1:6;
hence contradiction by A31, A46, EUCLID:57, EUCLID:58; :: thesis: verum
end;
( ( p2 `2 <= p2 `1 & - (p2 `1 ) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) <= (- (p2 `1 )) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) ) ) by A31, A34, XREAL_1:66;
then A47: ( ( p2 `2 <= p2 `1 & (- (p2 `1 )) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) <= (p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))) ) or ( |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 >= |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 & |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 <= - (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ) ) by A32, A33, A34, XREAL_1:66;
then ( ( |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 <= |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 & - (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) <= |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) or ( |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 >= |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 & |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 <= - (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ) ) by A32, A33, A34, XREAL_1:66;
then A48: Sq_Circ . |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| = |[((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 )))),((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))))]| by A41, JGRAPH_2:11, JGRAPH_3:def 1;
A49: (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) = p2 `1 by A32, A34, A36, XCMPLX_1:90;
A50: (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) = p2 `2 by A33, A34, A36, XCMPLX_1:90;
A51: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def 1;
not |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 = 0 by A32, A33, A34, A41, A47, XREAL_1:66;
then A52: (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 > 0 by SQUARE_1:74;
A53: (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 >= 0 by XREAL_1:65;
(((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) / ((sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) ^2 )) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 )))) ^2 ) > 1 by A36, A37, A38, A39, A40, XCMPLX_1:77;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) / ((sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) ^2 )) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) / ((sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) ^2 )) > 1 by XCMPLX_1:77;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) / ((sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) ^2 )) > 1 by A35, SQUARE_1:def 4;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) > 1 by A35, SQUARE_1:def 4;
then ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 )))) * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 )) > 1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 )) by A35, XREAL_1:70;
then ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) + ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) > 1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 )) ;
then ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) + ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 ))) > 1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 )) by A35, XCMPLX_1:88;
then A54: ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) > 1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 )) by A35, XCMPLX_1:88;
1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 )) ^2 )) = 1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) / ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 )) by XCMPLX_1:77;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 )) - 1 > (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) / ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ))) - 1 by A54, XREAL_1:11;
then ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 )) - 1) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) > (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) / ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 )) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) by A52, XREAL_1:70;
then A55: (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) - 1)) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) > (|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 by A52, XCMPLX_1:88;
((((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 )) + ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 )) - (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) * 1))) - ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) = (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) - 1) * (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 )) ;
then ( ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) - 1 > 0 or ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `2 ) ^2 ) < 0 ) by A55, XREAL_1:52;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 ) ^2 ) - 1) + 1 > 0 + 1 by A52, A53, XREAL_1:8;
then ( |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 > 1 ^2 or |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| `1 < - 1 ) by SQUARE_1:119;
then |[((p2 `1 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `2 ) / (p2 `1 )) ^2 ))))]| in Kb by A1;
hence ex x being set st
( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A29, A48, A49, A50, A51, EUCLID:57; :: thesis: verum
end;
case A56: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1 ) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1 ) ) ) ; :: thesis: ex x being set st
( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x )
set px = |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]|;
A57: ( ( p2 `1 <= p2 `2 & - (p2 `2 ) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2 ) ) ) by A56, JGRAPH_2:23;
A58: |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 = (p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) by EUCLID:56;
A59: |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 = (p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) by EUCLID:56;
1 + (((p2 `1 ) / (p2 `2 )) ^2 ) > 0 by XREAL_1:36, XREAL_1:65;
then A60: sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )) > 0 by SQUARE_1:93;
A61: 1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ) > 0 by XREAL_1:36, XREAL_1:65;
A62: (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) = (p2 `1 ) / (p2 `2 ) by A58, A59, A60, XCMPLX_1:92;
A63: p2 `2 = ((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) / (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) by A60, XCMPLX_1:90
.= (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) / (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) by EUCLID:56 ;
A64: p2 `1 = ((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))) / (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) by A60, XCMPLX_1:90
.= (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) by EUCLID:56 ;
A65: |.p2.| ^2 = ((p2 `2 ) ^2 ) + ((p2 `1 ) ^2 ) by JGRAPH_3:10;
A66: |.p2.| ^2 > 1 ^2 by A30, SQUARE_1:78;
A67: now
assume that
A68: |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 = 0 and
A69: |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 = 0 ; :: thesis: contradiction
A70: (p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) = 0 by A68, EUCLID:56;
(p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) = 0 by A69, EUCLID:56;
then p2 `1 = 0 by A60, XCMPLX_1:6;
hence contradiction by A56, A60, A70, XCMPLX_1:6; :: thesis: verum
end;
( ( p2 `1 <= p2 `2 & - (p2 `2 ) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) <= (- (p2 `2 )) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) ) ) by A57, A60, XREAL_1:66;
then A71: ( ( p2 `1 <= p2 `2 & (- (p2 `2 )) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) <= (p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))) ) or ( |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 >= |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 & |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 <= - (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ) ) by A58, A59, A60, XREAL_1:66;
then ( ( |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 <= |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 & - (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) <= |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) or ( |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 >= |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 & |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 <= - (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ) ) by A58, A59, A60, XREAL_1:66;
then A72: Sq_Circ . |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| = |[((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 )))),((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))))]| by A67, JGRAPH_2:11, JGRAPH_3:14;
A73: (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) = p2 `2 by A58, A60, A62, XCMPLX_1:90;
A74: (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) = p2 `1 by A59, A60, A62, XCMPLX_1:90;
A75: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def 1;
not |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 = 0 by A58, A59, A60, A67, A71, XREAL_1:66;
then A76: (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 > 0 by SQUARE_1:74;
A77: (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 >= 0 by XREAL_1:65;
(((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) / ((sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) ^2 )) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 )))) ^2 ) > 1 by A62, A63, A64, A65, A66, XCMPLX_1:77;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) / ((sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) ^2 )) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) / ((sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) ^2 )) > 1 by XCMPLX_1:77;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) / ((sqrt (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) ^2 )) > 1 by A61, SQUARE_1:def 4;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) > 1 by A61, SQUARE_1:def 4;
then ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 )))) * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 )) > 1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 )) by A61, XREAL_1:70;
then ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) + ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) > 1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 )) ;
then ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) + ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) / (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 ))) > 1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 )) by A61, XCMPLX_1:88;
then A78: ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) > 1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 )) by A61, XCMPLX_1:88;
1 * (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) / (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 )) ^2 )) = 1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) / ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 )) by XCMPLX_1:77;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 )) - 1 > (1 + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) / ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ))) - 1 by A78, XREAL_1:11;
then ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 )) - 1) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) > (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) / ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 )) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) by A76, XREAL_1:70;
then A79: (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) + (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) - 1)) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) > (|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 by A76, XCMPLX_1:88;
((((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 )) + ((((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) * ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 )) - (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) * 1))) - ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) = (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) - 1) * (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 )) ;
then ( ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) - 1 > 0 or ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `1 ) ^2 ) + ((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) < 0 ) by A79, XREAL_1:52;
then (((|[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 ) ^2 ) - 1) + 1 > 0 + 1 by A76, A77, XREAL_1:8;
then ( |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 > 1 ^2 or |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| `2 < - 1 ) by SQUARE_1:119;
then |[((p2 `1 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 )))),((p2 `2 ) * (sqrt (1 + (((p2 `1 ) / (p2 `2 )) ^2 ))))]| in Kb by A1;
hence ex x being set st
( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A29, A72, A73, A74, A75, EUCLID:57; :: thesis: verum
end;
end;
end;
hence y in Sq_Circ .: Kb by FUNCT_1:def 12; :: thesis: verum